Question: The lifespans of seals in a particular zoo are normally distributed. The average seal lives $14$ years; the standard deviation is $2.2$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a seal living between $9.6$ and $11.8$ years.
Answer: $14$ $11.8$ $16.2$ $9.6$ $18.4$ $7.4$ $20.6$ $95\%$ $68\%$ $13.5\%$ $13.5\%$ We know the lifespans are normally distributed with an average lifespan of $14$ years. We know the standard deviation is $2.2$ years, so one standard deviation below the mean is $11.8$ years and one standard deviation above the mean is $16.2$ years. Two standard deviations below the mean is $9.6$ years and two standard deviations above the mean is $18.4$ years. Three standard deviations below the mean is $7.4$ years and three standard deviations above the mean is $20.6$ years. We are interested in the probability of a seal living between $9.6$ and $11.8$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $95\%$ of the seals will have lifespans within 2 standard deviations of the average lifespan. It also tells us that $68\%$ of the seals will have lifespans within 1 standard deviation of the mean. The probability of a particular seal living between $9.6$ and $11.8$ years is $\color{orange}{13.5\%}$.